Do you mean "similar statements" for the sum of divisors function $\sigma(n)=\sum_{d\mid n}d$? Because there are plenty of other multiplicative function for which similar asymptotics are known.
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Wadim ZudilinJun 15 '10 at 11:29

I'm sorry for not being clear enough, it's my first question here, though. I mean similar statments for the $\sigma(n)$ function, not necessarily asymptotics, but anything that involves limit points of the function $\frac{\sigma(n)}{n \log \log n}$. For example, is there an important sequence $a_n$ such that $\frac{\sigma{a_n}}{n \log \log n}$ converges, besides the sequence of primes? A result that establishes the connection between the density and the limit superior? Etc. Nothing particular.
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nikmilJun 15 '10 at 22:53

1 Answer
1

where the limit of the Choie, Lichiardopol, Moree and Sole's
$$f_1(a_n) = \frac{\sigma(a_n)}{a_n \log \log a_n}$$
is the same
$$ e^\gamma .$$
That is, the limit for these numbers is the lim sup for all numbers.

These are more natural than people realize. There is a simple recipe that takes some $ \epsilon > 0$ and gives an explicit factorization for the best value $n_\epsilon;$ see page 7 in the Briggs pdf
"Notes on the Riemann hypothesis and abundant numbers" at the bottom of the Wikipedia entry. The exponent of a prime $p$ in the factorization of $n_\epsilon$ is
$$ \left\lfloor \log_p \left( \frac{p^{1 + \epsilon} - 1}{p^\epsilon -1} \right) \right\rfloor - 1 $$

The process of making a sequence of "champion" numbers this way was invented by Ramanujan.